Nonparametric regression using smoothing splines
Smoothing is fitting a smooth curve to data in a scatterplot
Will focus on two variables:
Y
and one
X
Our model:
y
i
=
f
(
x
i
) +
ε
i
,
where
ε
1
, ε
1
, . . . ε
n
are independent with mean 0
f
is some unknown smooth function
Up to now
f
has a specified form with unknown parameters
f
could be linear or nonlinear in the parameters,
functional form always specified
If
f
not determined by the subject matter, we may prefer to let the
data suggest a function form
c
2011 Dept. Statistics (Iowa State University)
Stat 511 section 30
1 / 50
Why estimate
f
?
can see features of the relationship between
X
and
Y
that are
obscured by error variation
summarizes the relationship between
X
and
Y
provide a diagnostic for a presumed parametric form
Example: Diabetes data set in Hastie and Tibshirani’s book
Generalized Additive Models
Examine relationship between age of diagnosis of diabetes and
log of the serum Cpeptide concentration
Here’s what happens if we fit increasing orders of polynomial, then
fit an estimated
f
c
2011 Dept. Statistics (Iowa State University)
Stat 511 section 30
2 / 50
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Age at Diagnosis
log Cpeptide concentration
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2011 Dept. Statistics (Iowa State University)
Stat 511 section 30
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5
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15
3.0
4.0
5.0
6.0
Age at Diagnosis
log Cpeptide concentration
linear fit
c
2011 Dept. Statistics (Iowa State University)
Stat 511 section 30
4 / 50
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 Spring '08
 Staff
 Regression Analysis, STATE UNIVERSITY

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